Portfolio Optimization

Transcription

1 Portfolio Optimization Jaehyun Park Ahmed Bou-Rabee Stephen Boyd EE103 Stanford University November 22, 2014

2 Outline Single asset investment Portfolio investment Portfolio optimization Single asset investment 2

3 Return of an asset over one period asset can be stock, bond, real estate, commodity,... invest in a single asset over period (quarter, week, day,...) buy q shares at price p (at beginning of investment period) h = pq is dollar value of holdings sell q shares at new price p + (at end of period) profit is qp + qp = q(p + p) = p+ p p h define return r = (p + p)/p return = investment profit profit = rh example: invest h = $1000 over period, r = +0.03: profit = $30 Single asset investment 3

4 Short positions basic idea: holdings h and share quantities q are negative called shorting or taking a short position on the asset (h or q positive is called a long position) how it works: you borrow q shares at the beginning of the period and sell them at price p at the end of the period, you have to buy q shares at price p + to return them to the lender all formulas still hold, e.g., profit = rh example: invest h = $1000, r = 0.05: profit = +$50 no limit to how much you can lose when you short assets normal people (and mutual funds) don t do this; hedge funds do Single asset investment 4

5 Return of an asset over multiple periods invest over periods t = 1,2,...,T (quarters, trading days, minutes, seconds...) p t is price at the beginning of period t return over period t is r t = pt+1 pt p t invest dollar amount h t in period t, or share number q t = h t /p t profit over period t is r t h t total profit is T t=1 r th t per period profit is 1 T T t=1 r th t Single asset investment 5

6 Examples stock prices of BP (BP) and Coca-Cola (KO) for last 10 years 70 KO BP Prices Days Single asset investment 6

7 Examples price changes over a few weeks 70 KO BP Prices Days Single asset investment 7

8 Examples returns of the assets over the same period KO BP Returns Days Single asset investment 8

9 Buy and hold a very simple choice of h t q t = q for all t = 1,2,...,T buy q shares at the beginning of period 1 sell q shares at the end of period T hence h t = p t q profit is T r t h t = t=1 T ( ) pt+1 p t (p t q) = q(p T+1 p 1 ) t=1 same as combining periods 1,...,T into a single period p t Single asset investment 9

10 Cumulative value plot plot of h t = p t q versus t (h 1 = $10,000 by tradition) 2.5 x 104 KO MSFT 2 Value Days Single asset investment 10

11 Constant value another simple choice of h t : h t = h, t = 1,...,T number of shares q t = h/p t (which varies with t) requires buying or selling shares every period to keep value constant profit is T t=1 r th per period profit is (1/T) T t=1 r th = avg(r t )h (in $) mean return is avg(r t ) (fractional; often expressed in %) profit standard deviation is std(r t h) = std(r t )h (in $) risk is std(r t ) (fractional) want per period profit high, risk low Single asset investment 11

12 Annualizing return and risk mean return and risk are often expressed in annualized form (i.e., per year) if there are P trading periods per year annualized return = P avg(r t) annualized risk = P std(r t) (the squareroot in risk annualization comes from the assumption that the fluctuations in return around the mean are independent) if t denotes trading days, with 250 trading days in a year annualized return = 250avg(r t) annualized risk = 250std(r t) Single asset investment 12

13 Risk-return plot annualized risk versus annualized return of various assets up (high return) and left (low risk) is good 25 SBUX 20 Annualized Return MMM GS BRCM 5 USDOLLAR Annualized Risk Single asset investment 13

14 Outline Single asset investment Portfolio investment Portfolio optimization Portfolio investment 14

15 Portfolio of assets n assets n-vector p t is prices of assets in period t, t = 1,2,...,T n-vector h t is dollar value holdings of the assets total portfolio value: V t = 1 T h t n-vector q t is the number of shares: (q t ) i = (h t ) i /(p t ) i w t = (1/1 T h t )h t gives portfolio weights or allocation (fraction of portfolio, defined only for 1 T h t > 0) Portfolio investment 15

16 Examples (h 3 ) 5 = 1000 means you short asset 5 in investment period 3 by $1,000 (w 2 ) 4 = 0.20 means 20% of total portfolio value in period 2 is invested in asset 4 w t = (1/n,...,1/n), t = 1,...,T means total portfolio value is equally allocated across assets in all investment periods 1 T h t = 0 means total short positions = total long positions Portfolio investment 16

17 Buy and hold portfolio same idea as single asset case (n-vector) q t = q for all t = 1,...,T holdings given by (h t ) i = (p t ) i q i, i = 1,...,n profit is T rt T h t = t=1 T n t=1 i=1 (r t ) i ( (pt+1 ) i (p t ) i (p t ) i ) ((p t ) i q i ) = q T (p T+1 p 1 ) Portfolio investment 17

18 Constant value portfolio simple choice of h t : h t = h, t = 1,...,T requires rebalancing (buying and selling) shares to maintain (h t ) i = h i every period profit is T t=1 rt t h per period profit is (1/T) T t=1 rt t h (in $) mean return is avg(r T t h)/1 T h (fractional; often expressed in %) profit standard deviation is std(r T t h) (in $) risk is std(r T t h)/1 T h (fractional) Portfolio investment 18

19 Risk-return plot for constant value portfolio good Annualized Return uniform bad Annualized Risk Portfolio investment 19

20 Constant weight portfolio with re-investment fix weight vector w given initial total investment V 1, set h 1 = V 1 w V 2 = V 1 +r1 T h 1 set h 2 = V 2 w, i.e., re-invest total portfolio value using allocation w and so on... V T = V 1 (1+r1 T w)(1+r2 T w) (1+rT Tw) V t 0 (or some small value like 0.1V 1 ) called going bust or ruin Portfolio investment 20

21 Cumulative value plot uniform portfolio between two assets, with h 1 = $10,000 (by tradition) 3 x 104 uniform portfolio individual assets Value Days Portfolio investment 21

22 Cumulative value plot portfolio with large short positions (heavily leveraged) going bust (dropping to 10% of starting value) 3 x 104 leveraged portfolio individual assets Value Days Portfolio investment 22

23 Comparison: Re-investment or not constant value portfolio (without re-investment) gives total profit T t=1 rt t h constant weight portfolio (with re-investment) gives total profit V T V 1 = ((1+r T Tw) (1+r T 1 w) 1)V 1 for r T t w all small (say, 0.01) so V T V 1 T t=1 rt t wv 1 (1+r T Tw) (1+r T 1 w) 1+ T rt T w t=1 profit with constant value h profit with constant weight w = (1/1 T h)h and initial investment h Portfolio investment 23

24 Comparison: Re-investment or not x 104 no reinvestment reinvestment Value Days Portfolio investment 24

25 Outline Single asset investment Portfolio investment Portfolio optimization Portfolio optimization 25

26 Portfolio optimization how should we choose a portfolio value vector h, or a portfolio weight vector w, over some investment period? more generally, these vectors could change with time, as our information or goals changes when we choose h or w, we know past returns ( realized returns ) but (of course) not future ones in all cases, we want high (mean) return, low risk Portfolio optimization 26

27 Returns matrix define returns matrix R = r T 1... r T T jth column is asset j return time series Rh is profit time series 1 T Rh is total profit avg(rh) = (1/T)1 T Rh is per period profit std(rh) is per period risk goal: choose h that makes avg(rh) high, std(rh) low Portfolio optimization 27

28 Portfolio optimization via least squares on past returns minimize std(rh) 2 = (1/T) Rh ρb1 2 subject to 1 T h = B, avg(rh) = ρb h is holdings vector to be found R is the returns matrix for past returns Rh is the (past) profit time series require mean (past) profit ρb minimize the standard deviation of (past) profit we are really asking what would have been the best constant allocation, had we known future returns Portfolio optimization 28

29 Constant weight portfolio optimization minimize std(rw) 2 = (1/T) Rw ρ1 2 subject to 1 T w = 1, avg(rw) = ρ very similar to constant weight optimization (in fact the two solutions are the same, except for scaling) w is weight allocation vector to be found Rw is the (past) return time series require mean (past) return ρ minimize the standard deviation of (past) return Portfolio optimization 29

30 Examples optimal w for annual return 1% (last asset is risk-less with 1% return) w = (0.0000,0.0000,0.0000,...,0.0000,0.0000,1.0000) optimal w for annual return 13% w = (0.0250, , ,..., ,0.0633,0.5595) optimal w for annual return 25% w = (0.0500, , ,..., ,0.1265,0.1191) asking for higher annual return yields more invested in risky, but high return assets larger short positions ( leveraging ) Portfolio optimization 30

31 Cumulative value plots for optimal portfolios cumulative value plot for optimal portfolios and some individual assets optimal portfolio, rho=0.20/250 optimal portfolio, rho=0.25/250 individual assets 10 5 Value Days Portfolio optimization 31

32 Optimal risk-return curve red curve obtained by solving problem for various values of ρ Annualized Return Annualized Risk Portfolio optimization 32

33 Optimal portfolios perform significantly better than individual assets risk-return curve forms a straight line one end of the line is the risk-free asset two-fund theorem: optimal portfolio w is an affine function in ρ w ν 1 ν 2 = R T R 1 R T 1 1 T T R RT 1 1 ρt Portfolio optimization 33

34 The big assumption now we make the big assumption (BA): future returns will look something like past ones you are warned this is false, every time you invest it is often reasonably true in periods of market shift it s much less true if BA holds (even approximately), then a good weight vector for past (realized) returns should be good for future (unknown) returns for example: choose w based on last 2 years of returns then use w for next 6 months Portfolio optimization 34

35 Optimal risk-return curve trained on 900 days (red), tested on the next 200 days (blue) here BA held reasonably well 25 Train Test 20 Annualized Return Annualized Risk Portfolio optimization 35

36 Optimal risk-return curve corresponding train and test periods 5 x Train Test Portfolio optimization 36

37 Optimal risk-return curve and here BA didn t hold so well (can you guess when this was?) 25 Train Test Annualized Return Annualized Risk Portfolio optimization 37

38 Optimal risk-return curve corresponding train and test periods 5 x Train Test Portfolio optimization 38

39 Rolling portfolio optimization for each period t, find weight w t using L past returns r t 1,...,r t L variations: update w every K periods (say, monthly or quarterly) add cost term κ w t w t 1 2 to objective to discourage turnover, reduce transaction cost add logic to detect when the future is likely to not look like the past add signals that predict future returns of assets (...and pretty soon you have a quantitative hedge fund) Portfolio optimization 39

40 Rolling portfolio optimization example cumulative value plot for different target returns update w daily, using L = 400 past returns 1.3 x rho=0.05/250 rho=0.1/250 rho=0.15/ Value Days Portfolio optimization 40

41 Rolling portfolio optimization example same as previous example, but update w every quarter (60 periods) 1.3 x rho=0.05/250 rho=0.1/250 rho=0.15/ Value Days Portfolio optimization 41